Make chickpea patties: In a 3- to 4-qt. pan, mix chickpea flour and dry seasonings.

Gradually whisk in 2 1/2 cups water until smooth. Cook over medium heat, whisking often, until mixture begins to thicken, 3 to 5 minutes. Switch to a spoon and stir until mixture pulls away from pan bottom and mounds in center of pan, 4 to 6 minutes.

Stir in parsley. Using an oiled spoon or fingers, evenly spread mixture in an oiled 9- by 13-in. pan. Let cool.

Make mayo: Mix all ingredients together in a small bowl. Chill, covered.

Make slaw: Whisk together vinegar, oil, salt, and pepper in a medium bowl, then gently stir in carrots and radishes.

Now for the sandwiches! Preheat broiler and set a rack 4 inches from heat.

Split rolls, leaving one side attached, and open up. Place rolls cut side up in a large rimmed baking pan; lightly brush tops with 1 tablespoons olive oil. Broil until golden, about 1 minute. Tip onto counter and drape with foil.